However, I cannot conclusively prove that sup A does not exist at all for this set A. Intuitively I can see that the elements' difference is becoming smaller and smaller and smaller, hence they should eventually be limited to some specific value. Altho it says if max A exists, it's also the sup A, but it does not say if max A doesn't exist, then there is no sup A.

What am I missing?

Thanks in advance.

EDIT:
Quick and dirty - If I assumed sup A = 0.23, for example, would it be sufficient evidence that sup A does not exist if I show there is a value 0.223 which is still greater than all the elements in the set A, but lesser than the supposed 0.23. Then it follows that I can suppose sup A = 0.222223, which still satisfies the upper bound criteria, however is not the least of the upper bounds. How can I show that there is no sup A in this case?

However, I cannot conclusively prove that sup A does not exist at all for this set A. Intuitively I can see that the elements' difference is becoming smaller and smaller and smaller, hence they should eventually be limited to some specific value. Altho it says if max A exists, it's also the sup A, but it does not say if max A doesn't exist, then there is no sup A.

What am I missing?

Thanks in advance.

EDIT:
Quick and dirty - If I assumed sup A = 0.23, for example, would it be sufficient evidence that sup A does not exist if I show there is a value 0.223 which is still greater than all the elements in the set A, but lesser than the supposed 0.23. Then it follows that I can suppose sup A = 0.222223, which still satisfies the upper bound criteria, however is not the least of the upper bounds. How can I show that there is no sup A in this case?

The least upper bound axiom states that if [itex]A \subset \mathbb{R}[/itex] is bounded above then it has a supremum [itex]M[/itex]. If it happens that [itex]M \in A[/itex] then [itex]M[/itex] is the maximum of [itex]A[/itex], but it may be that [itex]M \notin A[/itex] in which case [itex]A[/itex] has no maximum.

To your example: try summing the geometric series [tex]
a_N = \sum_{n=1}^N \frac{2}{10^n} = 2 \sum_{n=1}^N \frac{1}{10^n}[/tex] for fixed [itex]N[/itex]. Then consider what happens if you make [itex]N[/itex] arbitrarily large.

However, I cannot conclusively prove that sup A does not exist at all for this set A. Intuitively I can see that the elements' difference is becoming smaller and smaller and smaller, hence they should eventually be limited to some specific value. Altho it says if max A exists, it's also the sup A, but it does not say if max A doesn't exist, then there is no sup A.

What am I missing?

Thanks in advance.

EDIT:
Quick and dirty - If I assumed sup A = 0.23, for example, would it be sufficient evidence that sup A does not exist if I show there is a value 0.223 which is still greater than all the elements in the set A, but lesser than the supposed 0.23. Then it follows that I can suppose sup A = 0.222223, which still satisfies the upper bound criteria, however is not the least of the upper bounds. How can I show that there is no sup A in this case?

Of course there IS a sup in this case; I don't know why you think otherwise. In fact, it is a property of real numbers that any bounded set of real numbers has a supremum. However, maximum and supremum need not be the same thing. In this case the set has no maximum, but that does not matter for the problem at hand.